Theorem onintexmid 4715

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)

Assertion

Ref	Expression
onintexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3868	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2		prmg 3830	. . . . . . 7 ⊢ (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
3	2	adantr 276	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4		zfpair2 4342	. . . . . . 7 ⊢ {𝑢, 𝑣} ∈ V
5		sseq1 3271	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑢, 𝑣}))
7	6	exbidv 1878	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥 ∈ 𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
8	5, 7	anbi12d 477	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9		inteq 3968	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → ∩ 𝑦 = ∩ {𝑢, 𝑣})
10		id 19	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
11	9, 10	eleq12d 2309	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
12	8, 11	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13		onintexmid.onint	. . . . . . 7 ⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)
14	4, 12, 13	vtocl 2877	. . . . . 6 ⊢ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
15	1, 3, 14	syl2anc 415	. . . . 5 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16		elpri 3728	. . . . 5 ⊢ (∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣} → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
17	15, 16	syl 14	. . . 4 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
18		incom 3421	. . . . . . 7 ⊢ (𝑣 ∩ 𝑢) = (𝑢 ∩ 𝑣)
19	18	eqeq1i 2246	. . . . . 6 ⊢ ((𝑣 ∩ 𝑢) = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
20		dfss1 3435	. . . . . 6 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑣 ∩ 𝑢) = 𝑢)
21		vex 2824	. . . . . . . 8 ⊢ 𝑢 ∈ V
22		vex 2824	. . . . . . . 8 ⊢ 𝑣 ∈ V
23	21, 22	intpr 3997	. . . . . . 7 ⊢ ∩ {𝑢, 𝑣} = (𝑢 ∩ 𝑣)
24	23	eqeq1i 2246	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
25	19, 20, 24	3bitr4ri 213	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ 𝑢 ⊆ 𝑣)
26	23	eqeq1i 2246	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ (𝑢 ∩ 𝑣) = 𝑣)
27		dfss1 3435	. . . . . 6 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑣)
28	26, 27	bitr4i 187	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ 𝑣 ⊆ 𝑢)
29	25, 28	orbi12i 776	. . . 4 ⊢ ((∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣) ↔ (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
30	17, 29	sylib 122	. . 3 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
31	30	rgen2a 2604	. 2 ⊢ ∀𝑢 ∈ On ∀𝑣 ∈ On (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
32	31	ordtri2or2exmid 4713	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ∩ cin 3219   ⊆ wss 3220  {cpr 3706  ∩ cint 3965  Oncon0 4503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by: (None)